Abstract
A state on a tripartite quantum system H_A⊗ H_B⊗H_C forms a Markov chain, i.e., quantum conditional independence, if it can be reconstructed from its marginal on H_A⊗ H_B by a quantum operation from H_B to H_B⊗H_C via the famous Petz map: a quantum Markov chain ρ_ABC satisfies ρ_ABC=ρ_BC^1/2(ρ_B^-1/2ρ_ABρ_B^-1/2⊗ id_C)ρ_BC^1/2. In this paper, we study the robustness of the Petz map for different metrics, i.e., the closeness of marginals implies the closeness of the Petz map outcomes. The robustness results are dimension-independent for infidelity δ and trace distance ε. The applications of robustness results are The sample complexity of quantum Markov chain tomography, i.e., how many copies of an unknown quantum Markov chain are necessary and sufficient to determine the state, is Θ((d_A²+d_C²)d_B²/δ), and Θ((d_A²+d_C²)d_B²/ε²) . The sample complexity of quantum Markov Chain certification, i.e., to certify whether a tripartite state equals a fixed given quantum Markov Chain σ_ABC or at least δ-far from σ_ABC, is Θ((d_A+d_C)d_B/δ), and Θ((d_A+d_C)d_B/ε²). O(min{d_Ad_B³d_C³,d_A³d_B³d_C}/ε²) copies to test whether ρ_ABC is a quantum Markov Chain or ε-far from its Petz recovery state. We generalized the tomography results into multipartite quantum system by showing O(n²max_i {d_i²d_i+1²}/δ) copies for infidelity δ are enough for n-partite quantum Markov chain tomography with d_i being the dimension of the i-th subsystem.